[FOM] Inconsistent Systems

[Rob Arthan]

On 12 Sep 2013, at 13:57, Harvey Friedman <hmflogic at gmail.com> wrote:

> …

> What axioms and rules of classical logic are appropriate here to use with CA so that we don't derive all formulas?
> 
> To address this question, it seems best to first remove absurdity entirely from CA and understand what is going on. CA without any absurdity will be denoted by CA(no). We will use the usual classical introduction and elimination rules for and, or, if then, and construe iff as an abbreviation. 
> 
> Let alpha be the sentence (forall x,y)(x in y). 
> 
> We argue in CA(no). Let A = {x: x in x implies alpha). 
> 
> LEMMA 1. CA(no) proves: A in A implies alpha.
> 
> LEMMA 2. CA(no) proves: A in A.
> 
> LEMMA 3. CA(no) proves: alpha. 
> 
> LEMMA 4. CA(no) proves every formula in the language of CA(no). 
> 
> I leave it to the experts whether this is convincing, whether this is new, and what implications it has for various foundational and philosophical enterprises? And what are the next things to look at?
> 
I am certainly not an expert (on almost anything), but do have two comments:

1) Your sentence alpha acts like absurdity since, given CA. we have "ex alpha quodlibet": alpha implies that every atomic formula is both true and false (for the latter note that alpha implies that for any x and y, x is in {z | z is not in y}, i.e., x is not in y).

2) The rule of contraction seems to be necessary for Russell's paradox: if R is {x | x not in x} and phi is the formula "R in R", the argument of the paradox deduces "not phi" from "phi implies not phi", i.e., from "phi and phi implies false". So there is an assumption that "phi" is as strong as "phi and phi". I have a recollection that this was one of the motivations for Lukasiewicz's multi-valued logic (In which "phi and phi" may be strictly stronger than "phi"). but I don't have a relevant reference to hand.

Regards,

Rob.